Robert Black wrote:
> Is there a constructively provable version of the Tychonov product
> theorem? And if there is, exactly what does it say, and what
> definition of 'compact' does it use?
Yes, see for example Steve Vicker's "Some constructive roads to
Tychonoff", published in "From Sets and Types to Topology and Analysis:
Towards Practicable Foundations for Constructive Mathematics", Edited by
Laura Crosilla and Peter Schuster, Oxford University Press, 2005,
http://www.oup.com/uk/catalogue/?ci=9780198566519
The paper can be downloaded at
http://www.cs.bham.ac.uk/~sjv/InfiniteTych.pdf
Best regards,
Andrej